In this paper we present a novel admission control scheme for a mixed set of periodic and aperiodic tasks with hard deadlines under EDF, which can achieve near optimal performance with practical utility. The proposed admission control scheme is based on a novel schedulability measure for a deadline-constrained task, called utilization demand, which can be viewed as a combination of the processor time demand and the utilization factor. We first show that this new schedulability measure provides a necessary and sufficient schedulability condition for aperiodic tasks. We then present an efficient schedulability test for a mixed set of periodic and aperiodic tasks under EDF. The resulting schedulability test can be implemented as an on-line admission control algorithm of O(n) complexity, which in practice incurs sufficiently low overhead. Our experimental results show that the proposed admission control scheme outperforms existing approaches with respect to achievable processor utilization.